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5
Hydrodynamic Pressure Evaluation of
Reservoir Subjected to Ground
Excitation Based on SBFEM
Shangming Li
Institute of Structural Mechanics, China Academy of Engineering Physics
Mianyang City, Sichuan Province
China
1. Introduction
Dynamic responses of dam-reservoir systems subjected to ground motions are often a major
concern in the design. To ensure that dams are adequately designed for, the hydrodynamic
pressure distribution along the dam-reservoir interface must be determined for assessment
of safety.
Due to the fact that analytical methods are not readily available for dam-reservoir systems
with arbitrary geometry shape, numerical methods are often used to analyze responses of
dam-reservoir systems. In numerical methods, dams are often discretized into solid finite
elements through Finite Element Method (FEM), while the reservoir is either directly
modeled by Boundary Element Method (BEM) or is divided into two parts: a near field with
arbitrary geometry shape and a far field with a uniform cross section. The near field is
discretized into acoustic fluid finite elements by using FEM or boundary elements by BEM,
while the far field is modeled by BEM or a Transmitting Boundary Condition (TBC). Based
on these numerical methods, several coupling procedures were developed.
A FEM-BEM coupling procedure was used to implement the linear and non-linear analysis
of dam-reservoir interaction problems (Tsai & Lee, 1987; Czygan & Von Estorff, 2002),
respectively, in which the dam was modeled by FEM, while the reservoir was modeled by
BEM. A BEM-TBC coupling method was adopted to solve dam-water-foundation
interaction problems and dam-reservoir-sediment-foundation interaction problems
(Dominguez & Maeso, 1993; Dominguez et al., 1997). The dam and the near field of the
reservoir were discretized by using BEM, while the far field of the reservoir was represented
by a TBC. As a traditional numerical method, BEM has been popular in simulating
unbounded medium, but it needs a fundamental solution and includes a singular integral,
which affect its application. In order to avoid deriving a fundamental solution required in
BEM, the TBC attracted some researchers’ interests. A Sommerfeld-type TBC was used to
represent the far field (Kucukarslan et al., 2005), while a Sharan-type TBC was proposed for
infinite fluid (Sharan, 1987). The Sommerfeld-type and Sharan-type TBCs are readily
implemented in FEM due to their conciseness, but a sufficiently large near field is required
to model accurately the damping effect of semi-infinite reservoir. Except for the
aforementioned TBCs, an exact TBC (Tsai & Lee, 1991), a novel TBC (Maity &
Hydrodynamics – Natural Water Bodies
90
Bhattacharyya, 1999) and a non-reflecting TBC (Gogoi & Maity, 2006) were proposed,
respectively. These complicated TBCs gave better results even when a small near field was
chosen, but their implementations in a finite element code became complex and tedious.
In this chapter, the scaled boundary finite element method (SBFEM) was chosen to model
the far field. The SBFEM does not require fundamental solutions and is able to model
accurately the damping effect of semi-infinite reservoir and incorporate with FEM readily,
but the SBFEM requires the geometry of far field is layered (or tapered). Although BEM and
some of TBCs can handle far fields with arbitrary geometry, far fields in most dam-reservoir
systems are always chosen to be layered with a uniform cross section, which ensures the
SBFEM can be used in dam-reservoir interaction problems.
Based on a mechanically-based derivation, the SBFEM was proposed for infinite medium
(Wolf & Song, 1996a; Song & Wolf, 1996) which was governed by a three-dimensional
scalar wave equation and a three-dimensional vector wave equation, respectively. A
dynamic stiffness matrix and a dynamic mass matrix were introduced to represent infinite
medium in the frequency domain and the time domain, respectively. The dynamic
stiffness matrix satisfies a non-linear ordinary differential equation of first order, while
the dynamic mass matrix is governed by an integral convolution equation. The SBFEM
reduces spatial dimensions by one. Only boundaries need discretization and its solutions
in the radial direction are analytical. Therefore, it can handle well bounded domain
problems with cracks and stress singularities and unbounded domain problems including
infinite soil or unbounded acoustic fluid medium. In analyzing crack and stress
singularities problems, the SBFEM placed the scaling center on the crack tip and only
discretized the boundary of bounded domain using supper elements except the straight
traction free crack faces, which permitted a rigorous representation of the stress
singularities around the crack tip (Song, 2004; Song & Wolf, 2002; Yang & Deeks, 2007).
The response of unbounded domain problems was obtained by using the SBFEM alone or
coupling FEM and the SBFEM. A FEM-SBFEM coupling procedure was used to analyze
unbounded soil-structure interaction problems in the time domain (Ekevid & Wiberg,
2002; Bazyar & Song, 2008). For unbounded acoustic fluid medium problems, a FEM-
SBFEM coupling procedure combined with acoustic approximations was proposed to
evaluate the responses of submerged structures subjected to underwater shock waves in
the time domain (Fan et al., 2005; Li & Fan, 2007). Results showed that the SBFEM was
able to model accurately the damping behavior of the unbounded soil and infinite
acoustic fluid medium, but it was computationally expensive because the evaluations of
the dynamic mass matrix and dynamic responses need solving integral convolution
equations. In the frequency domain, dynamic condensation and substructure deletion
methods were used to evaluate the dynamic stiffness matrix, which avoid evaluating
integral convolution equations, but evaluation errors increased with frequency increasing
so that results at high frequencies were not acceptable (Wolf & Song, 1996b). To evaluate
accurately high frequencies behaviors of the dynamic stiffness matrix, a Pade series was
presented to analyze out-of-plane motion of circular cavity embedded in full-plane
through using the SBFEM alone (Song & Bazyar, 2007). Good results were obtained at
high frequencies, but results at low frequencies were inferior even if a high order Pade
series was used. The high order Pade series was not only complex, and also increased
computational cost. A simplified SBFEM formulation was presented through discovering
a zero matrix and a FEM-SBFEM coupling procedure was used to analyze dam-reservoir
interaction problems subjected to ground motions (Fan & Li, 2008). The simplified SBFEM
Hydrodynamic Pressure Evaluation of
Reservoir Subjected to Ground Excitation Based on SBFEM
91
was well suitable for all frequencies and no additional computational costs were increased
for low frequency analysis in comparison with for high frequency analysis. Its advantages
were exhibited by analyzing the harmonic responses of dam-reservoir systems in the
frequency domain. However in the time domain, its advantages are not as obvious as
those in the frequency domain because integral convolutions still need evaluating.
Although a Riccati equation and Lyapunov equations were presented to solve the integral
convolutions (Wolf & Song, 1996b), solving them needed great computational costs,
especially for large-scale systems, which limited the SBFEM applications in the time
domain. To simplify the integral convolutions and save computational costs, some
recursive formulations were proposed (Paronesso & Wolf, 1998; Yan et al., 2004), based on
a diagonalization procedure and the linear system theory (Paronesso & Wolf, 1995). The
integral convolution was transformed into an equivalent system of linear equations,
named state-variable description which was represented by finite-difference equations.
However, the coefficient matrix quaternion of finite-difference equations was calculated
by using Hankel matrix realization algorithms, which complicated the analysis.
Furthermore, the diagonalization procedure increased the order of the dynamic mass
matrix, and some global lumped parameters, such as springs, dashpots and masses, used
in the diagonalization procedure must be introduced at additional internal nodes
corresponding to inner variables in the state-variable description, besides the nodes on the
structure-medium interface. The number of global lumped parameters would become
very large for large-scale systems. This weakened the feasibility of the diagonalization
procedure. A new diagonalization procedure of the SBFEM for semi-infinite reservoir was
proposed (Li, 2009), whose calculation efficiency was proven to be high, although it still
included convolution integrals. With the improvement of the SBFEM evaluation efficiency
in the time domain analysis, the SBFEM will show gradually its advantages and potential
to solve problems including unbounded soil or unbounded acoustic fluid medium, such
as the dam-reservoir interaction problems.
2. Problem statement
Consider dam-reservoir interaction problems subjected to horizontal ground accelerations.
The dam-reservoir system and its Cartesian coordinate system were shown in Fig.1. The
Fig. 1. Dam-reservoir system
Dam
Dam-reservoir
interface
Near
field
HL
Free surface
Reservoir
bottom
Near-far-field
interface
x
y
Far field
Hydrodynamics – Natural Water Bodies
92
dam was subjected to a horizontal ground acceleration
x
a and the semi-infinite reservoir
was filled with an inviscid isentropic fluid. To evaluate the response of the dam-reservoir
system under a horizontal ground acceleration
x
a
excitation, the semi-infinite reservoir was
divided into two parts: a near field and a far field. The near field was located between the
dam-reservoir interface and the radiation boundary (the near-far-field interface at xL ),
while the far field was from xL
to
. Note that the geometry of the reservoir was chosen
to be arbitrary for x 0
and flat for x 0 .
For an inviscid isentropic fluid (acoustic fluid) with the fluid particles undergoing only
small displacements and not including body force effects, the governing equations is
expressed as
c
2
2
1
(1)
where
denotes velocity potential and c denotes the sound speed in fluid. Reservoir
pressure
p
, the velocity vector v and the velocity potential
have a relationship as follows:
v
(2a)
p
(2b)
where
denotes fluid density. Boundary conditions of the near field for Eq.(1) are following.
Along the dam-reservoir interface, one has
n
v
n
vn
(3)
where the unit vector
n is perpendicular to the dam-reservoir interface and points outward
of fluid;
n
v
is the normal velocity of the dam-reservoir interface. The boundary condition
along the reservoir bottom is
n
qv
n
(4)
where
q
is defined as
r
r
q
c
1
1
1
(5)
in which
r
denotes a reflection coefficient of pressure striking the bottom of the reservoir.
By ignoring effects of surface waves of fluid, the boundary condition of the free surface is
taken as
0
(6)
The boundary condition on the radiation boundary (near-far-filed interface) should include
effects of the radiation damping of infinite reservoir and those of energy dissipation in the
reservoir due to the absorptive reservoir bottom. To model these effects accurately, the
SBFEM was adopted in this chapter.
Hydrodynamic Pressure Evaluation of
Reservoir Subjected to Ground Excitation Based on SBFEM
93
3. SBFEM formulation
Fig.2 showed the SBFEM discretization model of the far field shown in Fig.1, which was a
layered semi-infinite fluid medium whose scaling center was located at minus infinity. The
whole semi-infinite layered far field was divided into some layered sub-fields. Each layered
sub-field was represented by one element on the near-far-field interface, so the whole far
field was discretized into some elements on the near-far-field interface. Based on the
discretization, a dynamic stiffness or mass matrix was introduced to describe the
characteristics of the far field in the SBFEM. The interaction between the near field and the
far field was expressed as the following SBFEM formulation.
Fig. 2. SBFEM discretization model of layered far field
3.1 SBFEM formulation in the frequency domain
On the discretized near-far-field interface, the SBFEM formulation in the frequency domain
(Fan & Li, 2008; Li et al., 2008) for the far field filled with unbounded acoustic fluid medium
is written as
n
VSΦ
(7)
where
Φ denotes the column vector composed of nodal velocity potentials
;
S is
the dynamic stiffness matrix of the far field and
n
V
satisfies
e
w
Te
n
f
nw
e
vd
VN
(8)
in which
n
v is the normal velocity;
w
denotes the near-far-field interface;
f
N is the shape
function for a typical discretized acoustic fluid finite element; and
e
denotes an
assemblage of all fluid elements on the near-far-field interface. The dynamic stiffness matrix
S (Li, et al., 2008) satisfies
T
ii
2
101 1 2 0 0
SEESEEC M0
(9)
L
H
Near-far-field interface
x
y
Layered sub-fields
Reservoir botto
m
Free surface
Hydrodynamics – Natural Water Bodies
94
where global coefficient matrices
0
E
,
1
E
,
2
E
,
0
C and
0
M only depend on the geometry of
the near-far-field interface and the reflection coefficient
r
. They are obtained through
assembling all elements’
e
0
E ,
e
1
E ,
e
2
E ,
e
0
C and
e
0
M on the near-far-field interface. The
matrices
e
0
E
,
e
1
E
,
e
2
E
,
e
0
C
and
e
0
M
corresponding to each element can be evaluated
numerically or analytically using the following equations.
T
e
dd
11
011
11
EBBJ
(10a)
T
e
dd
11
121
11
EBBJ
(10b)
T
e
dd
11
222
11
EBBJ
(10c)
T
eff
dd
c
11
0
2
11
1
MNNJ
(10d)
where the
f
N is defined in Eq.(8) and the others
J
,
1
B ,
2
B are defined below. The matrix
J
is defined as
00
fff
fff
H
ddd
ddd
ddd
ddd
NNN
J
xyz
NNN
xyz
(11a)
where the symbol
H denotes the water depth in the far field and x , y and z are element
nodal coordinates column vectors. Due to the fact that x coordinates of all nodes inside the
near-far-field interface (vertical surface) are same, the matrix
J
becomes
00
0
0
ff
ff
H
dd
dd
dd
dd
NN
J
yz
NN
yz
(11b)
Write the inverse of
J
in the following form
jjj
jjj
jjj
11 12 13
1
21 22 23
31 32 33
J
(12)
The components
mn
j
mn, 1,2,3 can be evaluated by using Eq.(11b). Therefore, the
matrix
1
B
is defined as
Hydrodynamic Pressure Evaluation of
Reservoir Subjected to Ground Excitation Based on SBFEM
95
f
j
j
j
11
1
21
31
BN
(13)
and the matrix
2
B is
ff
jj
dd
jj
dd
jj
12 13
2
22 23
32 33
NN
B
(14)
Note that Eqs.(10-14) are only the functions of nodal coordinates of elements inside the near-
far-field interface. The matrix
e
0
C
is a zero matrix for elements not adjacent to reservoir bottom
inside the near-far-field interface, while for those adjacent to reservoir bottom,
e
0
C satisfies
b
T
r
e
ff
b
r
Hd
c
0
1
1
1
CNN
(15)
where the symbol
b
denotes the reservoir bottom of the near-far-field interface, i.e. the line
y 0
as shown in the Fig.2. Assembling all elements’
e
0
E ,
e
1
E ,
e
2
E ,
e
0
C and
e
0
M can yield
the global coefficient matrices
0
E ,
1
E ,
2
E ,
0
C and
0
M in Eq.(9). Details about them can be
found in the literatures (Wolf & Song, 1996b; Li et al., 2008).
For a vertical near-far-field interface as shown in Fig.2, as the matrix
1
E was a zero matrix,
the dynamic stiffness matrix
S in Eq.(9) can be re-written readily as
i
2020010
SECMEE
(16)
where
is an excitation frequency. The
S
can be obtained by the Schur factorization.
3.2 SBFEM formulation in the time domain
The corresponding SBFEM formulation of Eq.(7) in the time domain is written as (Wolf &
Song, 1996b)
t
n
ttd
0
VMΦ
(17)
in which
t
M is the dynamic mass matrix of the far field;
tΦ
and
n
tV
are the
corresponding variables of
Φ and
n
V in the time domain, respectively.
t
M and
i
2
S forms a Fourier transform pair. Upon discretization of Eq.(17) with respect to
time and assuming all initial conditions equal to zero, one can get the following equation
n
j
nn
nnjnj
j
1
11
1
VMΦ MMΦ
(18)
in which
nj
nj t
1
1
MM ,
j
jt
ΦΦ and
n
nn
nt
VV where t denotes
an increment in time step.
Applying the inverse Fourier transformation to Eq. (9) with
1
0
E yields
Hydrodynamics – Natural Water Bodies
96
t
tt
td t
32
200
0
0
62
mm ecm
(19)
where
t is time and
T
tt
11
mUMU
(20)
T2121
eUEU
(21)
T0101
mUMU (22)
T0101
cUCU (23)
in which U satisfies
T0
EUU
(24)
A procedure (Wolf & Song, 1996b) was presented to evaluate the dynamic mass matrix
t
M at different time t governed by the convolution integral Eq.(19). In that procedure,
discretization of Eq.(19) with respect to time was implemented, and an algebraic Riccati
equation for evaluating
tt
M at first time step and a Lyapunov equation for
evaluating
tjt
M at other jth time steps were formed, respectively. The
tjt
M
at any time was obtained by utilizing Schur factorization to solve these two types of
equations. When the coefficient matrix
0
0
c
, a simple diagonal procedure (Li, 2009) can be
adopted to evaluate the
t
M , which can avoid Schur factorization and solving Riccati
equation and Lyapunov equation.
4. FEM-SBFEM coupling formulation of reservoir
To obtain the response of dam-reservoir system, the near-field fluid domain is discretized
into an assemblage of finite elements. The corresponding finite-element governing equation
of Eq.(1) for the near-field domain can be expressed as
n
n
n
11 12 13 1 11 12 13 1 1
21 22 23 2 21 22 23 2 2
31 32 33 3 31 32 33 3 3
mmm Φ kkk Φ V
mmm Φ kkk Φ V
mmm Φ kkk Φ V
(25)
where the global mass matrix
m , the global stiffness matrix k and the global vector
n
V are
treated in the standard manner as in the traditional FE procedures; the subscripts 1 and 2
refer to nodal variables at the dam-reservoir interface and the near-far-field interface,
respectively, while the subscript 3 refers to other interior nodal variables in the near-field
fluid. At the near-far-field interface, the near-field FEM-domain couples with the far-field
SBFEM-domain. The kinematic continuity condition requires that both fields have the same
normal velocity at the near-far-field interface. Hence, one has
nn2
VV
(26)